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3) Let D = {(x, y) E R² | 0 < x² + y² < 4}. Give all boundary points of D. Is D open, closed or bounded? Justify all your answers. The definition of open set is in your Ebook in section 13.2. The related definitions of closed and bounded set are as follows: Closed: A set D is closed if it contains all of its boundary points. Bounded: A subset D of IR" is bounded if it is contained in some open ball D,(0). For example the interval [-1,5) is neither open nor closed since it contains some but not all of its endpoints. It is bounded since it is contained in the open interval D6 (0)=(-6,6).

3) Let D = {(x, y) E R² | 0 < x² + y² < 4}. Give all boundary points of D. Is D open, closed or bounded? Justify all your answers. The definition of open set is in your Ebook in section 13.2. The related definitions of closed and bounded set are as follows: Closed: A set D is closed if it contains all of its boundary points. Bounded: A subset D of IR" is bounded if it is contained in some open ball D,(0). For example the interval [-1,5) is neither open nor closed since it contains some but not all of its endpoints. It is bounded since it is contained in the open interval D6 (0)=(-6,6).

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 3** Consider the set \( D = \{(x, y) \in \mathbb{R}^2 \mid 0 < x^2 + y^2 \leq 4\} \). Determine all boundary points of \( D \). Is \( D \) open, closed, or bounded? Justify all your answers. The definition of an open set can be found in your Ebook in section 13.2. The related definitions of closed and bounded sets are as follows: **Closed:** A set \( D \) is closed if it contains all of its boundary points. **Bounded:** A subset \( D \) of \( \mathbb{R}^n \) is bounded if it is contained in some open ball \( D_r(0) \). For instance, the interval \([-1, 5)\) is neither open nor closed since it contains some but not all of its endpoints. However, it is bounded since it is contained in the open interval \( D_6(0) = (-6, 6) \).
Transcribed Image Text:**Problem 3** Consider the set \( D = \{(x, y) \in \mathbb{R}^2 \mid 0 < x^2 + y^2 \leq 4\} \). Determine all boundary points of \( D \). Is \( D \) open, closed, or bounded? Justify all your answers. The definition of an open set can be found in your Ebook in section 13.2. The related definitions of closed and bounded sets are as follows: **Closed:** A set \( D \) is closed if it contains all of its boundary points. **Bounded:** A subset \( D \) of \( \mathbb{R}^n \) is bounded if it is contained in some open ball \( D_r(0) \). For instance, the interval \([-1, 5)\) is neither open nor closed since it contains some but not all of its endpoints. However, it is bounded since it is contained in the open interval \( D_6(0) = (-6, 6) \).
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